Contributions from conjugacy classes of regular elliptic elements in Hermitian modular groups to the dimension formula of Hermitian modular cusp forms
Author:
Min King Eie
Journal:
Trans. Amer. Math. Soc. 294 (1986), 635645
MSC:
Primary 11F46; Secondary 11F55, 11F72, 32N15
MathSciNet review:
825727
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Abstract: The dimension of the vector space of hermitian modular cusp forms on the hermitian upper half plane can be obtained from the Selberg trace formula; in this paper we shall compute the contributions from conjugacy classes of regular elliptic elements in hermitian modular groups by constructing an orthonomal basis in a certain Hilbert space of holomorphic functions. A generalization of the main Theorem can be applied to the dimension formula of cusp forms of . A similar theorem was given for the case of regular elliptic elements of in [5] via a different method.
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 Hel Braun, Hermitian modular functions, Ann. of Math. (2) 50 (1949), 827855. MR 0032699 (11:333a)
 [2]
 , Hermitian modular functions. III. The Hermitian modular group, Ann. of Math. (2) 53 (1951), 143180. MR 0039005 (12:482c)
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 Minking Eie, Dimension formulas for the vector spaces of Siegel's modular cusp forms of degree two and degree three, Thesis, University of Chicago, 1982, pp. 1246.
 [4]
 , Dimensions of spaces of Siegel cusp forms of degree two and three, Mem. Amer. Math. Soc. No. 304 (1984), pp. 1185. MR 749684 (86c:11036)
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 , Contributions from conjugacy classes of regular elliptic elements in to the dimension formula, Trans. Amer. Math. Soc. 285 (1984), 403410. MR 748846 (86c:11037)
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 R. Godement, Généralités sur les formes modulaires. I, II, Séminaire Henri Cartan, 10e années, 1957, 1958.
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 L. K. Hua, On the theory of functions of several complex variables. I, II, III, Amer. Math. Soc. Transl. 32 (1962), 163263.
 [8]
 , Inequalities involving determinants, Amer. Math. Soc. Transl. 32 (1962), 265272.
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 Suehiro Kato, A dimension formula for a certain space of automorphic forms of , Math. Ann. 266 (1984), 457477. MR 735528 (86h:11045)
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 Hans Maass, Siegel's modular forms and Dirichlet series, Lecture Notes in Math., vol. 216, SpringerVerlag, Berlin and New York, 1971. MR 0344198 (49:8938)
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 S. Helgason, Differential geometry and symmetric spaces, Academic Press, New York, 1962. MR 0145455 (26:2986)
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 George W. Machkey, Unitary group representation in physics, probability and number theory, Benjamin, New York, 1978.
 [13]
 A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. 20 (1956), 4787. MR 0088511 (19:531g)
 [14]
 Hideo Shimizu, On discontinuous groups operating on the product of the upper half plane, Math. Ann. 177 (1963), 3371. MR 0145106 (26:2641)
 [15]
 C. L. Siegel, Lectures on quadratic forms, Tata Institute of Fundamental Research, Bombay, 1967. MR 0271028 (42:5911)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198608257276
PII:
S 00029947(1986)08257276
Article copyright:
© Copyright 1986 American Mathematical Society
